Rewriting supremum over an image set

Question

Let $h\colon X \to \mathbb{R}$ be a function on a (separable) Banach space $X$. Consider $$\sup_{x \in f(C)} h(x)$$

where $f:Y \to X$ is a map and $C$ is some given subset of $Y$, another separable Banach space. When is this supremum equal to $$\sup_{c \in C} h(f(c))?$$

Isn't it equal always? If $x \in f(C)$, then $x=f(c)$ for some $c$. Can't we just replace it?

Answer 1

Yes, these are always equal. The notation $\sup_{x \in f(C)} h(x)$ really means the supremum of the set $\{h(x):x\in f(C)\}$, which is equal to the set $\{h(f(c)):c\in C\}$. So both of your suprema are the suprema of the same set.